We consider large scale, distributed storage systems with a redundancy mechanism; cloud storage being a prime example. We investigate how such systems can reduce their power consumption during low-utilization time intervals by operating in a low-power mode. In a low power mode, a subset of the disks or nodes are powered down, yet we ask that each data item remains accessible in the system; this is called full coverage. The objective is to incorporate this option into an existing system rather than redesign the system. When doing so, it is crucial that the low power option should not affect the performance or other important characteristics of the system during full-power (normal) operation. This work is a comprehensive study of what can or cannot be achieved with respect to full coverage low power modes. The paper addresses this question for generic distributed storage systems (where the key component under investigation is the placement function of the system) as well as for specific popular system designs in the realm of storing data in the cloud. Our observations and techniques are instrumental for a wide spectrum of systems, ranging from distributed storage systems for the enterprise to cloud data services. In the cloud environment where low cost is imperative, the effects of such savings are magnified by the large scale.